Observationally, the X-ray spectrum (0.5 − 10 keV) of low-level accreting neutron stars (NSs) (L 0.5−10keV 10 36 erg s −1 ) can generally be well fitted by the model with two components, i.e, a thermal soft X-ray component plus a power-law component. Meanwhile, the fractional contribution of the power-law luminosity η (η ≡ L power law 0.5−10keV /L 0.5−10keV ) varies with the X-ray luminosity L 0.5−10keV . In this paper, we systematically investigate the origin of such X-ray emission within the framework of the advection-dominated accretion flow (ADAF) around a weakly magnetized NS, in which the thermal soft X-ray component arises from the surface of the NS and the power-law component arises from the ADAF itself. We test the effects of the viscosity parameter α in the ADAF and thermalized parameter f th (describing the fraction of the ADAF energy released at the surface of the NS as thermal emission) on the relation of η versus L 0.5−10keV . It is found that η is nearly a constant (∼ zero) with L 0.5−10keV for different α with f th = 1, which is inconsistent with observations. Meanwhile, it is found that a change of f th can significantly change the relation of η versus L 0.5−10keV . By comparing with a sample of non-pulsating NS-low mass X-ray binaries probably dominated by low-level accretion onto NSs, it is found that a small value of f th 0.1 is needed to match the observed range of η 10% in the diagram of η versus L 0.5−10keV . Finally, we argue that the small value of f th 0.1 implies that the radiative efficiency of NSs with an ADAF accretion may not be as high as the predicted result previously of ǫ ∼Ṁ GM R * /Ṁc 2 ∼ 0.2 despite the existence of the hard surface.